Pathologies of the large- N limit for RP^N-1 , CP^N-1 , QP^N-1 and mixed isovector/isotensor σ -models

We compute the phase diagram in the N→∞ limit for lattice RP^N-1, CP^N-1 and QP^N-1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.